论文标题
常规稀疏抗魔法正方形的结构
Constructions of regular sparse anti-magic squares
论文作者
论文摘要
图形标记是图理论中众所周知且经过深入研究的问题。稀疏的抗魔法正方形可用于构建图形顶点魔术标记。对于积极的整数$ n，d $和$ d <n $，a $ n \ times n $ array $ a $ a $基于$ \ {0,1，\ cdots，nd \} $称为\ emph {用密度$ d $}稀疏的反魔术$ n $ of conders $ n $}，sams $（n，n，d）$（n，d） $ \ {1,2，\ cdots，nd \} $正好出现$ a $的一个条目，其行-SUMS，列和两个主对角和两个构成了一组$ 2N+2 $连续的整数。如果在每行，每一列和每个主角的$ d $正分段中完全存在$ d $正面条目，则为SAMS $（n，d）$称为\ emph {常规}。在本文中，我们研究了定期稀疏的抗魔法正方形$ n \ equiv1,5 \ pmod 6 $，并且证明对于任何$ n \ equiv1,5 \ pmod 6 $，存在常规的SAMS $（n，d）$，并且只有$ 2 \ leq D \ leq D \ leq d \ leq leq n-1 $。
Graph labeling is a well-known and intensively investigated problem in graph theory. Sparse anti-magic squares are useful in constructing vertex-magic labeling for graphs. For positive integers $n,d$ and $d<n$, an $n\times n$ array $A$ based on $\{0,1,\cdots,nd\}$ is called \emph{a sparse anti-magic square of order $n$ with density $d$}, denoted by SAMS$(n,d)$, if each element of $\{1,2,\cdots,nd\}$ occurs exactly one entry of $A$, and its row-sums, column-sums and two main diagonal sums constitute a set of $2n+2$ consecutive integers. An SAMS$(n,d)$ is called \emph{regular} if there are exactly $d$ positive entries in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares of order $n\equiv1,5\pmod 6$, and it is proved that for any $n\equiv1,5\pmod 6$, there exists a regular SAMS$(n,d)$ if and only if $2\leq d\leq n-1$.
